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          第一章 误差
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        <h2 id="误差分类">1.误差分类</h2>
<h3 id="绝对误差和绝对误差限">1.1.绝对误差和绝对误差限</h3>
<p>定义：<span class="math inline">\(e^*=x-x^*\)</span>，<span class="math inline">\(x\)</span>表示真实值，一般是未知的，<span class="math inline">\(x^*\)</span>表示预测值。<span class="math inline">\(|e^*|&lt;\varepsilon^*\)</span>，<span class="math inline">\(\varepsilon^*\)</span>称为绝对误差限。</p>
<h3 id="相对误差和相对误差限">1.2.相对误差和相对误差限</h3>
<p>定义：<span class="math inline">\(e^*_r=\frac{e^*}{x}\)</span>，由于真实的<span class="math inline">\(x\)</span>未知，所以使用<span class="math inline">\(x^*\)</span>代替<span class="math inline">\(x\)</span>。<span class="math inline">\(|e^*_r|&lt;\varepsilon^*_r\)</span>，<span class="math inline">\(\varepsilon^*_r\)</span>称为相对误差限。</p>
<a id="more"></a>
<h2 id="有效数字和有效位数">2.有效数字和有效位数</h2>
<p><span class="math inline">\(|\pi-3.1416|&lt;0.5\times10^{-4}\)</span>，<strong>有效位数</strong>是5，3.1416是<strong>有效数字</strong><br />
有效数字的浮点表示： <span class="math display">\[
x^* = 10^{m}\times (0.a_1a_2\cdots a_n)\times 10^{-n}\\
a_1\in \{0, 1, \cdots, 9\},a_i\in\{1, \cdots, 9\},i\in\{1,\cdots, n\}\\
\]</span> 这里m称为阶，一共n位有效数字，<span class="math inline">\(a_1,a_2\cdots,a_n\)</span>是有效数字。<span class="math inline">\(e^*&lt;\frac{1}{2}\times 10^{m-n}\)</span>，是绝对误差限。</p>
<h3 id="说明">2.1说明</h3>
<ol type="1">
<li>四舍五入的时候，必须从有效数字开始，例如3.142有四位有效数字，四舍五入为3.14，3.141有三位有效数字，四舍五入为3.1。<br />
</li>
<li>准确值有无限的有效位数，<span class="math inline">\(\varepsilon^*\rightarrow 0\)</span>。<br />
</li>
<li>有效数字相同，绝对误差不一定相同。</li>
</ol>
<h3 id="相对误差限和有效位数的关系">2.2.相对误差限和有效位数的关系</h3>
<p><span class="math display">\[
x^* = 10^{m}\times (0.a_1a_2\cdots a_n)\times 10^{-n}
\]</span> 则相对误差限为：<span class="math inline">\(\varepsilon_r^*&lt;\frac{1}{2a_1}\times 10^{-(n-1)}\)</span><br />
相反的，若相对误差限为：<span class="math inline">\(\varepsilon_r^*&lt;\frac{1}{2(a_1+1)}\times 10^{-(n-1)}\)</span>，则至少有n位有效数字。</p>
<h2 id="数值运算的误差估计">3.数值运算的误差估计</h2>
<h3 id="四则运算的绝对误差限">3.1.四则运算的绝对误差限</h3>
<p><span class="math display">\[
\begin{aligned}
    \varepsilon (x_1^*+x_2^*)&amp;\approx \varepsilon (x_1^*) + \varepsilon (x_2^*)\\
    \varepsilon (x_1^*x_2^*)&amp;\approx |x_1^*|\varepsilon (x_2^*) + |x_2^*|\varepsilon (x_1^*)\\  
    \varepsilon (\frac{x_1^*}{x_2^*})&amp;\approx \frac{|x_1^*|\varepsilon (x_2^*) + |x_2^*|\varepsilon (x_1^*)}{|x_2^*|^2}
\end{aligned}
\]</span></p>
<h3 id="函数的误差估计">3.2.函数的误差估计</h3>
<p>在<span class="math inline">\(x^*\)</span>处Taylor展开有： <span class="math display">\[
\begin{aligned}
    f(x)&amp;=f(x^*)+f^1(x^*)(x-x^*)+\frac{1}{2!}f^2(x^*)(x-x^*)^2\\
    \Rightarrow f(x)-f(x^*)&amp;=f^1(x^*)(x-x^*)，舍去二次项
\end{aligned}
\]</span></p>
<ul>
<li><p>一元函数绝对误差 <span class="math display">\[
\begin{aligned}
f(x)-f(x^*)&amp;=f^1(x^*)(x-x^*)\\
e_z^*&amp;=f^1(x^*)e_x^*
\end{aligned}
\]</span></p></li>
<li><p>二元函数绝对误差 <span class="math display">\[
\begin{aligned}
f(x,y)-f(x^*, y^*)&amp;=f_x^1(x^*, y^*)(x-x^*)+f_y^1(x^*, y^*)(y-y^*)\\
e_z^*&amp;=f_x^1(x^*, y^*)e_x^*+f_y^1(x^*, y^*)e_y^*
\end{aligned}
\]</span></p></li>
<li><p>多元函数绝对误差和相对误差 <span class="math display">\[
\begin{aligned}
e_z^*&amp;=f^1_{x_1}(x-x_1)+f^1_{x_2}(x-x_2)+\cdots+f^1_{x_n}(x-x_n)\\
e_{zr}^*&amp;=\frac{f^1_{x_1}(x-x_1)+f^1_{x_2}(x-x_2)+\cdots+f^1_{x_n}(x-x_n)}{A^*}
\end{aligned}
\]</span></p></li>
</ul>
<h2 id="代码编写原则">4.代码编写原则</h2>
<ol type="1">
<li>数值公式稳定</li>
<li>避免相近数相减</li>
<li>防止大数吃小数</li>
<li>绝对值小的数不宜做分母</li>
<li>简化计算步骤</li>
</ol>

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